CSS

Moving into the 21st century now, carbon capture and storage (CSS) is the newest technique pioneered in 2000 by Weyburn.  It's a method of CDR and technologies are currently being developed and applied in 74 countries on a global scale, displayed rather nicely on this interactive map.

CSS stops CO2 being released into the atmosphere at industrial sites by capturing it before it's released.  It is then transported via large pipelines (the cheapest way) and stored in geology, deep oceans or in minerals.  I'll dwell on the storage options a bit because I think it's interesting
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Storage in geology
The CO2 is injected into underground formations and be prevented from escaping by physical and geochemical trapping mechanisms:

Diagram illustrating the possible areas of carbon storage underground (Source: Global CSS Institute).

• injections into oil fields increases the amount of crude oil extraction - this technique is done often in the USA (IPCC 2010).
• In unmineable coal seams, CO2 molecules stick to the coals surface and is absorbed.  However, this displaces some of the methane the coal contains, which would either have to be sold, or burned which would offset the positive storage effects.
• Saline water saturated reservoirs are common, have a large area for storage and are of no use to humans.  They are currently used in some places to store chemical waste, but more research needs to be done in terms of trapping and leakage as relatively little is known about them in comparison to oil fields or coal seams.

 Potential sites for geological storage (IPCC special report)

Storage in oceans
CO2 can be transported via ships or pipelines to an ocean storage site deep below the thermocline, where it will remain for 100-1000 years out of contact with the atmosphere.

Some ways of ocean injection and storage (IPCC full report)

• Ship or pipeline injections at 1000-3000m below the surface, forming plumes or lakes.
• Plumes either rise or sink depending on and concentration.  Rising plumes causes CO2 to dissolve in the water column.  Sinking plumes are more dense and so sink along and inclined seafloor.  
• Carbon lakes are reservoirs created at injection depths greater than 3000m. 
• Alternatives including combining CO2 with a carbonate such as limestone and depositing it in the deep ocean, or adding carbon to existing clathrates. 

Ocean storage is probably the most controversial as it will inevitably increase acidity and cause detrimental effects to ecology, though the extent to which it will cause harm is poorly understood and requires further research.  Additionally, no large scale implementation of ocean storage has taken place to date, only small scale pilot studies.  The phenomena of ocean mixing will also mean that isolated carbon lakes will eventually reach surface waters and then return to the atmosphere via oceanic-atmospheric carbon exchange.

Mineral storage
Chemical reactions between metal oxides in silicate rocks and CO2 produce carbonates and silica which lock away carbon potentially forever.  Ideal materials would be those such as magnesium oxide or calcium oxide; which are naturally abundant, but oxides can also be found in industrial waste.  Once the materials are carbonised they will retain the carbon and not leak it back to the atmosphere, making this technique superior in terms of storage.  Additionally, this will mean the storage areas will not need to be monitored.  The diagram below summarises the technique:

(IPCC)
So it can be reused if it isn't disposed which is unique for CSS.


Potential problems
- High energy requirements
- Cost (although I think relatively cheap compared to other things I've looked at)
- Leakage, especially with ocean storage
- Negative ecological consequences (eg. ocean acidification)
- Carbon may enter groundwater supplies

Overall I think this is a really good idea! CO2 could potentially be stored in geology for millions of years (IPCC), with a 99% likelihood of them retaining the injected Carbon, and if mineral storage is employed it will remain there forever and never escape.  Oceans cover 70% of the Earth so they also hold huge potential to keep carbon locked away if buried deep enough.
Humans are less directly affected by offshore burial; with geological storage there is that 1% chance of leakage into soil or groundwater but due to cycling in the ocean acidification is likely to eventually occur; causing a threat to biodiversity.
These reports are really detailed and have everything you'll need to know ever on CSS:
IPCC summary
Full report